Now, let M be a n-dimensional differentiable manifold and let T⁢M be its tangent bundle.
The set of differentiable sections Γ⁢(M)={X:M→T⁢M} is a differentiable Lie algebra which is endowed with a differentiable inner productg:Γ⁢(M)×Γ⁢(M)→ℝ via

Now, if one uses a coordinated patch in M one has a set of n-coordinated vector fields ∂1,..,∂n
meaning ∂i=∂∂⁡ui being ui the coordinatefunctions.
These are also dubbed holonomic derivations.

So it makes sense to speak about the derivatives∇∂i⁡∂j
and since the ∂i are tangent which generate at a point Tp⁢(M), then ∇∂i⁡∂j
is also tangent, so there are n×n numbers (functions if one varies position) Γi⁢js which enters
in the relation

For a proof please see the last part in:
http://planetmath.org/?op=getobj&from=collab&id=64http://planetmath.org/?op=getobj&from=collab&id=64

Connection with base vectors.

Let us assume that coordinates ui are referred to a right-handedorthogonal Cartesian system with attached constant base vectors 𝐞i≡𝐞i and coordinates wj referred to a general curvilinear system attached to a local covariant base vectors 𝐠j and local contravariant base vectors 𝐠k, both systems embedded in the Euclidean spaceℝn. We shall also suppose diffeomorphic the transfomation ui↦wj. Then, by definition

comma denoting differentiation with respect to the curvilinear coordinateswj and g=|gj⁢k|. When the coordinate curves are orthogonal we have the following formulae for the Christoffel symbols: (repeated indices are not to be summed)